Key Terms

additive decomposition

time series decomposition into the sum of its components

augmented Dickey-Fuller (ADF) test

statistical test for stationarity of a time series

autocorrelation

measure of correlation between a time series and a shifted copy of itself

autoregressive (AR) model

model or component of a model that captures how the time series depends on its own previous values

cyclic component

large variations in the data that recur over longer time periods than seasonal fluctuations, having no fixed frequency

detrending

one of two complementary operations that separate the trend component from a time series

differencing

found by taking differences of consecutive terms, that is, $x_{n+1}-x_n$

error

extent to which predictions differ from actual observations

exponential moving average (EMA)

type of weighted moving average in which the most recent values are given larger weights and the weights decay exponentially the further back in time the values are in the series

forecasting

making predictions about future, unknown values of a time series

integrative (I) component

component of the ARIMA model that represents the differencing operation

lag

number of time steps that a time series is shifted for the purpose of computing autocorrelation

level

the mean of all the time series data

mean absolute error (MAE)

measure of error; $\text{MAE}=\frac{1}{n}\sum _{i=1}^n|{\epsilon }_i|$

mean absolute percentage error (MAPE)

measure of relative error; $\text{MAPE}=\frac{1}{n}\sum _{i=1}^n|\frac{{\epsilon }_i}{x_i}|$

measures of error (or measures of fit)

the metrics used to assess how well a model's predictions align with observed data

multiplicative decomposition

time series decomposition into the product of its components

naïve (or flat) forecasting method

using only the last observed value of a time series to predicting the next term

noise (or random error)

random, unpredictable variations that occur in the time series that cannot be attributed to the trend, cycles, or seasonal components

normalized (weights)

weights or coefficients that add up to a total of 1

order (or degree)

the number of terms or lags used in a model to describe the time series; for a sequence that is modeled by a polynomial formula, the value of the exponent on the term of highest order of the polynomial formula that models the sequence

peak

when computing ACF, a correlation value that is significantly larger than the previous values

period

smallest time interval over which a seasonal variation pattern repeats

prediction interval

range of values the variable could take with some level of probability in a forecast

residuals

the difference between the given time series and the model $(x_n-{\hat{x}}_n)$ quantifying the error of the model, or noise (random variation) of the time series

root mean squared error (RMSE)

measure of error; $\text{RMSE}=\sqrt{\frac{1}{n}\sum _{i=1}^n\left({{\epsilon }_i}^2\right)}$

scale-dependent

regarding a measure or error that is in direct proportion with the data

seasonal component (seasonality)

variation in the data that occurs at fixed time periods

Seasonal-Trend decomposition using LOESS (STL)

a powerful tool that decomposes a time series into the trend-cycle, seasonal, and noise components

sequence

ordered list of numbers, measurements, or other data

simple moving average (SMA)

average (arithmetic mean) of a fixed number of consecutive data points in a time series

stationary

characterizing a time series in which the variance is relatively constant over time, an overall upward or downward trend cannot be found, and no seasonal patterns exist

symmetric mean absolute percentage error (sMAPE)

measure of relative error; $\text{sMAPE}=\frac{1}{n}\sum _{i=1}^n\frac{2|{\epsilon }_i|}{|x_i|+|{\hat{x}}_i|}$

terms

in mathematics, the individual values of a sequence

time series

data that has a time component, or an ordered sequence of data points

time series analysis

the examination of data points collected at specific time intervals, enabling the identification of trends, patterns, and seasonal variations crucial for making informed predictions and decisions

time series model

function, algorithm, or method for finding, approximating, or predicting the values of a given time series

trend

long-term direction of time series data in the absence of any other variation

trend curve (or trendline)

line or curve that models the trend of a time series

trend-cycle component

component of a time series that combines the trend and long-term cycles

volatility

displaying significant fluctuations from the mean, typically due to factors that are difficult to analyze or predict

weighted moving average (WMA)

moving average in which terms are given weights according to some formula or rule

white noise

time series data that has constant mean value close to zero, constant variance, and no correlation from one part of the series to another

window

term used to describe a fixed number of consecutive terms of a time series